A note on general sliding window processes. (English) Zbl 1300.60043

Alon, Noga; Feldheim, Ohad Noy

Electron. Commun. Probab. 19, Paper No. 66, 7 p. (2014).

Summary: Let \(f:\mathbb{R}^k\to[r]=\{1,2,\dots,r\}\) be a measurable function, and let \(\{U_i\}_{i\in\mathbb{N}}\) be a sequence of i.i.d. random variables. Consider the random process \(\{Z_i\}_{i \in \mathbb N}\) defined by \(Z_i=f(U_{i},\dots{},U_{i+k-1})\). We show that, for all \(q\), there is a positive probability, uniform in \(f\), that \(Z_1=Z_2=\dots=Z_q\). A continuous counterpart is that, if \(f:\mathbb R^k \to \mathbb R\), and \(U_i\) and \(Z_i\) are as before, then there is a positive probability, uniform in \(f\), for \(Z_1,\dots,Z_q\) to be monotone. We prove these theorems, give upper and lower bounds for this probability, and generalize to variables indexed on other lattices.
The proof is based on an application of combinatorial results from Ramsey theory to the realm of continuous probability.

MSC:

60G07	General theory of stochastic processes
60C05	Combinatorial probability

Keywords:

\(k\)-factor; \(d\)-dependent; de Bruijn; Ramsey